Eigenvalues and cycles of consecutive lengths

As the counterpart of classical theorems on cycles consecutive lengths due to Bondy and Bollobás in spectral graph theory, Nikiforov proposed following open problem 2008: What is maximum C $C$ such that for all positive ε < $\varepsilon \lt C$ sufficiently large n $n$ , every G $G$ order with radius ρ ( ) > ⌊ 2 4 ⌋ $\rho (G)\gt \sqrt{\lfloor \frac{{n}^{2}}{4}\rfloor }$ contains a cycle length ℓ $\ell $ each integer ∈ [ 3 − ] \in [3,(C-\varepsilon )n]$ . We prove ≥ 1 $C\ge \frac{1}{4}$ improving existing bounds. Besides several novel ideas, our proof technique partly inspired by recent research Ramsey numbers star versus even Allen, Łuczak, Polcyn, Zhang, aid powerful inequality. also derive an Erdős–Gallai-type edge number condition cycles, which may be independent interest.